FoxHfunctions in fractional diffusion

2007

We deal with the Cauchy problem for the space-time fractional diffusion-wave equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order alpha in (0,2] and skewness theta, and the first-order time derivative with a Caputo derivative of order beta in (0,2]. The fundamental solution is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. By using the Mellin transform, we provide a general representation of the solution in terms of Mellin-Barnes integrals in the complex plane, which allows us to extend the probability interpretation known for the standard diffusion equation to suitable ranges of the relevant parameters alpha and beta. We derive explicit formulae (convergent series and asymptotic expansions), which enable us to plot the corresponding spatial probability densities.

We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2] and skewness θ (|θ| ≤ min {α, 2 − α}), and the first-order time derivative with a Caputo derivative of order β ∈ (0, 2] . The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We review the particular cases of space-fractional diffusion {0 < α ≤ 2 , β = 1} , time-fractional diffusion {α = 2 , 0 < β ≤ 2} , and neutral-fractional diffusion {0 < α = β ≤ 2} , for which the fundamental solution can be interpreted as a spatial probability density function evolving in time. Then, by using the Mellin transform, we provide a general representation of the Green functions in terms of Mellin-Barnes integrals in the complex plane, which allows us to extend the probability interpretation to the ranges {0 < α ≤ 2} ∩ {0 < β ≤ 1} and {1 < β ≤ α ≤ 2}. Furthermore, from this representation we derive explicit formulae (convergent series and asymptotic expansions), which enable us to plot the spatial probability densities for different values of the relevant parameters α, θ, β .

Waves and Stability in Continuous Media - Proceedings of the 11th Conference on WASCOM 2001, 2002

The fundamental solutions (Green functions) for the Cauchy problems of the spacetime fractional diffusion equation are investigated with respect to their scaling and similarity properties, starting from their composite Fourier-Laplace representation. By using the Mellin transform, a general representation of the Green functions in terms of Mellin-Barnes integrals in the complex plane is presented, that allows us to obtain their computational form in the space-time domain and to analyse their probability interpretation.

2002

We deal with the Cauchy problem for the space-time fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the second-order space derivative with a Riesz-Feller derivative of order α ∈ (0, 2] and skewness θ (|θ| ≤ min {α, 2 − α}), and the first-order time derivative with a Caputo derivative of order β ∈ (0, 2]. The fundamental solution (Green function) for the Cauchy problem is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. We review the particular cases of space-fractional diffusion {0 < α ≤ 2 , β = 1} , time-fractional diffusion {α = 2 , 0 < β ≤ 2} , and neutral-fractional diffusion {0 < α = β ≤ 2} , for which the fundamental solution can be interpreted as a spatial probability density function evolving in time. Then, by using the Mellin transform, we provide a general representation of the Green functions in terms of Mellin-Barnes integrals in the complex plane, which allows us to extend the probability interpretation to the ranges {0 < α ≤ 2} ∩ {0 < β ≤ 1} and {1 < β ≤ α ≤ 2}. Furthermore, from this representation we derive explicit formulae (convergent series and asymptotic expansions), which enable us to plot the spatial probability densities for different values of the relevant parameters α, θ, β .

Applied Mathematics and Computation, 2003

We revisit the Cauchy problem for the time-fractional diffusion equation, which is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order b 2 ð0; 2. By using the Fourier-Laplace transforms the fundamentals solutions (Green functions) are shown to be high transcendental functions of the Wright-type that can be interpreted as spatial probability density functions evolving in time with similarity properties. We provide a general representation of these functions in terms of Mellin-Barnes integrals useful for numerical computation.

Mathematics, 2017

In this paper, some new properties of the fundamental solution to the multi-dimensional space-and time-fractional diffusion-wave equation are deduced. We start with the Mellin-Barnes representation of the fundamental solution that was derived in the previous publications of the author. The Mellin-Barnes integral is used to obtain two new representations of the fundamental solution in the form of the Mellin convolution of the special functions of the Wright type. Moreover, some new closed-form formulas for particular cases of the fundamental solution are derived. In particular, we solve the open problem of the representation of the fundamental solution to the two-dimensional neutral-fractional diffusion-wave equation in terms of the known special functions.

Journal of Computational and Applied Mathematics, 2007

The fundamental solution of the fractional diffusion equation of distributed order in time (usually adopted for modelling sub-diffusion processes) is obtained based on its Mellin-Barnes integral representation. Such solution is proved to be related via a Laplace-type integral to the Fox-Wright functions. A series expansion is also provided in order to point out the distribution of time-scales related to the distribution of the fractional orders. The results of the time fractional diffusion equation of a single order are also recalled and then re-obtained from the general theory.

The fundamental solution of the fractional diffusion equation of distributed order in time (usually adopted for modelling sub-diffusion processes) is obtained based on its Mellin-Barnes integral representation. Such solution is proved to be related via a Laplace-type integral to the Fox-Wright functions. A series expansion is also provided in order to point out the distribution of time-scales related to the distribution of the fractional orders. The results of the time fractional diffusion equation of a single order are also recalled and then re-obtained from the general theory.

SpringerPlus, 2016

The transfer of heat in skin tissue is mainly a heat conduction process, which is coupled to several additional complicated physiological process, including blood circulation, sweating, metabolic heat generation and sometimes heat dissipation via hair or fur above the skin surface (Ozisik 1985). Accurate description of the thermal interaction between vasculature and tissue is essential for the advancement of medical technology in treating fatal disease such as tumors and skin cancer. Mathematical model has been used significantly in the analysis of hyperthermia in treating tumors, cryosurgery, fatalplacental studies, and many other applications (Minkowycz et al. 2009). Fractals and fractional calculus have been used to improve the modelling accuracy of many phenomena in natural science. The most important advantage of using fractional calculus approach is due to its non-local property. This means that the next state of a system depends not only upon its current state but also upon all of its historical states. Many researchers worked on fractional partial differential equations and gave very important results. Mainardi et al. (2005) obtained the fundamental solution of Cauchy problem for the space-time fractional diffusion equation in terms of H-function, Langlands (2006) gave the solution of a modified fractional diffusion equation on an infinite domain, Salim and El-Kahlout (2009) discussed exact solution of time fractional advection dispersion equation with reaction term, Saxena et al. (2006) obtained solution of generalized fractional kinetic equation in terms of Mittag-Leffler function, Haubold et al. (2011a) obtained solution of a fractional reaction diffusion equation in closed form, Huang and Guo (2010) gave the fundamental solutions to a class of the time fractional

Integral Transforms and Special Functions, 2004

The fundamental solution (Green function) for the Cauchy problem of the space-time fractional diffusion equation is investigated with respect to its scaling and similarity properties, starting from its Fourier-Laplace representation. Then, by using the Mellin transform, a general representation of the Green function in terms of Mellin-Barnes integrals in the complex plane is derived. This allows us to obtain a suitable computational form of the Green function in the space-time domain and to analyse its probability interpretation.